tropical geometry

Introduction

In The Riemann Hypothesis for Curves over Finite Fields, we gave a rough outline of Andre Weil’s strategy to prove the analogue of the famous Riemann hypothesis for curves over finite fields. A rather natural question to ask would be, does this strategy give us any suggestions on how to take on the original Riemann hypothesis? We mentioned briefly in The Field with One Element that some mathematicians hope to find in the theory of the so-called “field with one element” something that will allow them to apply the ideas of Weil’s proof to the original Riemann hypothesis, by viewing the scheme as some kind of “curve” over the “field with one element”.

In this post we will consider something along similar lines, examining a kind of “space” to which we can apply an analogue of Weil’s strategy. This approach is due to the mathematicians Alain Connes and Caterina Consani, and makes use of the concepts of sites and toposes (see More Category Theory: The Grothendieck Topos and Even More Category Theory: The Elementary Topos). This is perhaps appropriate, since sites or toposes are often referred to as “generalized spaces”.

We recall from The Riemann Hypothesis for Curves over Finite Fields some aspects of Weil’s strategy. The object in consideration is a curve over a finite field . In order to write down the zeta function for , we need to count the number of points over , for every from to infinity. We can do this by counting the fixed points of powers of the Frobenius morphism. Explicitly this means taking intersection numbers of the diagonal and the divisor formed by integral linear combinations of powers of the Frobenius morphism on , where is an algebraic closure of (it is the direct limit of the directed system formed by all the ) and . The number of points of will be the same as the number of points of over . Throughout this post we should keep these steps of Weil’s strategy in mind.

In order to transfer this strategy of Weil to the original Riemann hypothesis, Connes and Consani construct the arithmetic site, meant to be the analogue of , and the scaling site, meant to be the analogue of . The intuition behind these constructions is that the points of the scaling site, which is the same as the points of the arithmetic site “over “, is the same as the points of the “adele class space” , which originally came up in earlier work of Connes where he constructed a quantum-mechanical system which gives Riemann’s prime-counting function (whose study provided the historical origin of the Riemann hypothesis), in the form of Weil’s “explicit formula”, as a quantum-mechanical trace formula! In essence this work restates the Riemann hypothesis in terms of mathematical language more commonly associated to physics, and is part of Connes’ pioneering work in noncommutative geometry, a new area of mathematics also closely related to physics, in particular quantum mechanics and quantum field theory. In the definition of the adele class space, refers to the ring of adeles of (see Adeles and Ideles), while refers to , where are the -adic integers, which can be defined as the inverse limit of the inverse system formed by .

The Arithmetic Site

We now proceed to discuss the arithmetic site. It is described as the pair , where a Grothendieck topos, which, as we may recall from More Category Theory: The Grothendieck Topos, is defined as a category equivalent to the category of sheaves on a site . In the case of , is the category with only one object, and whose morphisms correspond to the multiplicative monoid of nonzero natural numbers (we also use to denote this category, and to denote the category with one object and whose morphisms correspond to ), while is the indiscrete, or chaotic, Grothendieck topology, where all presheaves are also sheaves.

As part of the definition of the arithmetic site, we must also specify a structure sheaf. In this case is provided by , the semiring (a semiring is like a ring, but is only a monoid, and not a group, under the “addition” operation – a semiring is also sometimes called a “rig“, because it is a ring without the “n” – the negative elements, and the most common example is the natural numbers with the usual addition and multiplication) whose elements are just the integers, together with , but where the “addition” is provided by the “maximum” operation, and the “multiplication” is provided by the ordinary addition! With the arithmetic site thus defined, we denote it by .

We digress for a while to discuss the semiring , as well as the closely related semirings (defined similarly to , but with the real numbers instead of the integers), (whose elements are the positive real numbers, with the addition given by the maximum operation, and the multiplication given by the ordinary multiplication), and the so-called Boolean semifield (whose elements are and , with the addition again given by the maximum operation, and the multiplication again given by the ordinary multiplication). These semirings have origins in the area of mathematics known as tropical geometry, so named because one of its pioneers, Imre Simon, comes from Brazil, which is a tropical country. However, another source of inspiration is the work of the mathematical physicist Viktor Pavlovich Maslov in “semiclassical” quantum mechanics, where certain approximations could be made as the quantum mechanical systems being studied approached the classical limit. Maslov considered a “conjugated” addition

and this just happened to be the same as .

Going back to the arithmetic site, we now discuss its points. Recall from Even More Category Theory: The Elementary Topos that a point of a topos (we discussed elementary toposes in that post, but this also applies to Grothendieck toposes) is defined by a geometric morphism from the topos of sheaves of sets on the singleton set (the set with a single element) to the topos. This refers to a pair of adjoint functors such that the left-adjoint is left-exact (preserves finite limits). Therefore, for the arithmetic site, a point is given by such a pair and such that is left-exact. The point is also uniquely determined by the covariant functor where is the Yoneda embedding.

There is an equivalence of categories between the category of points of the arithmetic site and the category of totally ordered groups which are isomorphic to the nontrivial subgroups of and injective morphisms of ordered groups. For such an ordered group we therefore have a point . This gives us a correspondence with (where refers to the ring of finite adeles of , which is defined similarly to the ring of adeles of except that the infinite prime is not considered) because any such ordered group is of the form , the ordered group of all rational numbers such that , for some unique . We can also now describe the stalks of the structure sheaf at the point ; it is isomorphic to the semiring , with elements given by the set , addition given by the maximum operation, and multiplication given by the ordinary addition.

The arithmetic site is analogous to the curve over the finite field . As for the finite field , its analogue is given by the Boolean semifield mentioned earlier, which has “characteristic “, reminiscent of the field with one element (see The Field with One Element). Next we want to find the analogues of the algebraic closure , as well as the Frobenius morphism. The former is given by the semiring , which contains , while the latter is given by multiplicative group of the positive real numbers , as it is isomorphic to the group of automorphisms of that keep fixed.

But while we do know that the points of the arithmetic topos are given by geometric morphisms and determined by contravariant functors , what do we mean by its “points over“? A point of the arithmetic site “over ” refers to the pair , where as earlier, and (we recall that are the stalks of the structure sheaf ). The points of the arithmetic site over include its points “over “, which are what we discussed earlier, and mentioned to be in correspondence with . But in addition, there are also other points of the arithmetic site over which are in correspondence with , just as contains all of but also other elements. Altogether, the points of the arithmetic site over correspond to the disjoint union of and , which is , the adele class space as mentioned earlier.

There is a geometric morphism (here is defined similarly to , but with in place of ) uniquely determined by

which sends the single object of to the sheaf on , which we now describe. Let denote the set of all rational numbers such that is an element of , where is the adele with a for the -th component and for all other components. Then the sheaf can be described in terms of its stalks , which are given by , the positive part of , and , given by . The sections are given by the maps such that for finitely many .

The Scaling Site

Now that we have defined the arithmetic site, which is the analogue of , and the points of the arithmetic site over , which is the analogue of the points of over the algebraic closure , we now proceed to define the scaling site, which is the analogue of . The points of the scaling site are the same as the points of the arithmetic site over , which is analogous to the points of being the same as the points of over . But the importance of the scaling site lies in the fact that we can construct the analogue of a sheaf of rational functions on it, and a Riemann-Roch theorem, which, as we may recall from The Riemann Hypothesis for Curves over Finite Fields, it is also an important part of Weil’s proof.

The scaling site is once again given by a pair , where is a Grothendieck topos and is a structure sheaf, but both are quite sophisticated constructions compared to the arithmetic site. To describe the Grothendieck topos we recall that it must be a category equivalent to the category of sheaves on some site . Here is the category whose objects are given by bounded open intervals , including the empty interval , and whose morphisms are given by

and in the special case that is the empty interval , we have

.

The Grothendieck topology here is defined by the collection of all ordinary covers of for any object of the category :

Now we have to describe the structure sheaf. We start by considering , the structure sheaf of the arithmetic site. By “extension of scalars” from to we obtain the reduced semiring . This is not yet the structure sheaf , because the underlying category and Grothendieck topology for the scaling site is more complicated than the arithmetic site, and unlike the case for the arithmetic site, for the scaling site not every presheaf is a sheaf. So we must first “localize” , and this gives us the structure sheaf .

Let us describe in more detail. Let be a rank subgroup of . Then an element of is given by a Newton polygon, which is the convex hull of the union of finitely many quadrants , where and (a set is a convex set if it contains the line segment connecting any two of its points; the convex hull of a set is the smallest convex set that contains it). The Newton polygon is uniquely determined by the function

for . This correspondence gives us an isomorphism between and , the semiring of convex, piecewise affine, continuous functions on with slopes in and finitely many singularities, with the pointwise operations (function is a convex function if the points on and above its graph form a convex set).

Therefore, we can describe the sections of the structure sheaf , for any bounded open interval , as the set of all convex, piecewise affine, continuous functions from to with slopes in . We can also likewise describe the stalks of the structure sheaf – for a point associated to a rank 1 subgroup , the stalk is given by the semiring of germs of -valued, convex, piecewise affine, continuous functions with slope in . We also have points with “support “, corresponding to the points of the arithmetic site over . For such a point, the stalk is given by the semiring associated to the totally ordered group .

Now that we have decribed the Grothendieck topos and the structure sheaf , we describe the scaling site as being given by the pair , and we denote it by .

Our next task, now that we have described the arithmetic site and the scaling site, is to find the analogue of the Riemann-Roch theorem. We start by noting that we have a sheaf of semifields , defined by letting be the semifield of fractions of . For an element in the stalk of , we define its order as

where

for .

We let be the set of all points of the scaling site such that is isomorphic to . The are the analogues of the orbits of Frobenius. There is a topological isomorphism . It is worth noting that the expression is reminiscent of the Tate uniformization of an elliptic curve (which generalizes the idea that an elliptic curve over the complex numbers forms a lattice in the complex plane to other complete fields besides the complex numbers – see The Moduli Space of Elliptic Curves).

We have a pullback sheaf , which we denote suggestively by . It is the sheaf on whose sections are convex, piecewise affine, continuous functions with slopes in . We can consider the sheaf of quotients of and its global sections , which are piecewise affine, continuous functions with slopes in such that for all . Defining

we have the following property for any (recall that the zeroth cohomology group is defined as the space of global sections of ):

We now want to define the analogue of divisors on (see Divisors and the Picard Group). A divisor on is a section , mapping to , of the bundle of pairs , where is isomorphic to , and . We define the degree of a divisor as follows:

Given a point such that for some , we have a map . This gives us a canonical mapping

Given a divisor on , we define

We have and if and only if , for i.e. is a principal divisor.

We define the group as the quotient of the group of divisors of degree on by the group of principal divisors on . The group is isomorphic to , while the group of divisors on modulo the principal divisors is isomorphic to .

In order to state the analogue of Riemann-Roch theorem we need to define the following module over :

Given , we define

where is the slope of at . Then we have the following increasing filtration on :

This allows us to define the following notion of dimension for (here refers to what is known as the topological dimension or Lebesgue covering dimension, a notion of dimension defined in terms of refinements of open covers):

The analogue of the Riemann-Roch theorem is now given by the following:

S-Algebras

This concludes our discussion of the arithmetic site and the scaling site, but I would like to discuss one more related topic also being explored by Connes and Consani – the use of -algebras, which is closely related to the -sets we have already introduced in The Field with One Element. Both of these concepts have their origins in homotopy theory.

We recall from the short discussion at the end of The Riemann Hypothesis for Curves over Finite Fields that the Weil conjectures, which are Weil’s generalization of the Riemann hypothesis for curves over finite fields to varieties of higher dimension, were proven by making use of cohomology (in particular etale cohomology) to find the fixed points of the powers of the Frobenius morphism (the formula that gives us the fixed points of a map using cohomology is called the Lefschetz fixed point formula). Now, concepts such as monoids, semirings, and many others (including the mathematician Nikolai Durov’s approach to the field with one element, which he also uses to develop a new version of Arakelov geometry) are all subsumed under the concept of -algebras, and doing so allows us to make use of a cohomology theory called topological cyclic cohomology. Connes and Consani hope that topological cyclic cohomology will help prove the original Riemann hypothesis the way that etale cohomology helped prove the Weil conjectures. Let us discuss briefly the work of Connes and Consani on this topic.

We recall from The Field with One Element the definition of a -set (there also referred to as a -space). A -set is defined to be a covariant functor from the category , whose objects are pointed finite sets and whose morphisms are basepoint-preserving maps of finite sets, to the category of pointed sets. An -algebra is defined to be a -set together with an associative multiplication and a unit , where is the inclusion functor (also known as the sphere spectrum). An -algebra is a monoid in the symmetric monoidal category of -sets with the wedge product and the sphere spectrum.

Any monoid defines an -algebra via the following definition:

for any pointed finite set . Here is the smash product of and as pointed sets, with the basepoint for given by its zero element element. The maps are given by , for .

Similarly, any semiring defines an -algebra via the following definition:

for any pointed finite set . Here refers to the set of basepoint preserving maps from to . The maps are given by for , , and . The multiplication is given by for any and . The unit is given by for all in , where if , and otherwise.

Therefore we can see that the notion of -algebra subsumes the notions of monoids and semirings, and other notions such as that of “hyperrings“, which we leave to the references for the moment. Instead, we will discuss how -algebras are related to the approach of Durov to the field with one element and Arakelov geometry. As we mentioned in Arakelov Geometry, the main idea of the theory is to consider the “infinite prime” along with the other points of . We therefore define as . Let be the structure sheaf of . We want to extend this to a structure sheaf on , and to accomplish this we will use the functor from semirings to -algebras defined earlier. For any open set containing , we define

.

The notation is defined for the -algebra associated to the semiring as follows:

where in this particular case comes from the usual absolute value on . This becomes available to us because the sheaf is a subsheaf of the constant sheaf .

Given an Arakelov divisor on (in this context an Arakelov divisor is given by a pair , where is an ordinary divisor on and is a real number) we can define the following sheaf of -modules over :

where is the real number “coefficient” of , and means, for an -module (here the -algebra is constructed the same as , except there is no multiplication or unit) with seminorm such that for and ,

With such sheaves of -algebras on now constructed, the tools of topological cyclic cohomology can be applied to it. The theory of topological cyclic cohomology is left to the references for now, but will hopefully be discussed in future posts on this blog.

Conclusion

The approach of Connes and Consani, whether making use of the arithmetic site and the scaling site to apply Weil’s strategy to the original Riemann hypothesis, or making use of -algebras and topological cyclic cohomology in analogy with the proof of the Weil conjectures, is still currently facing several technical obstacles. In the former case, an intersection theory and a Riemann-Roch theorem on the square of the scaling site is yet to be constructed. In the latter, there is the problem of appropriate coefficients for the cohomology theory. There are already several proposed strategies for dealing with these obstacles. Such efforts, aside from aiming to prove the Riemann hypothesis, widens the scope of the mathematics that we have today, and, perhaps more importantly, uncovers more and more the mysterious geometry underlying the familiar everyday concept of numbers.